Detection of nonlinear resonances among gravity modes of slowly pulsating B stars: results from five iterative prewhitening strategies
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Astronomy & Astrophysics manuscript no. main ©ESO 2021
August 9, 2021
Detection of nonlinear resonances among gravity modes of slowly
pulsating B stars: results from five iterative prewhitening strategies
J. Van Beeck1 , D. M. Bowman1 , M. G. Pedersen2 , T. Van Reeth1 , T. Van Hoolst1, 3 , and C. Aerts1, 4, 5
1
Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
e-mail: jordan.vanbeeck@kuleuven.be
2
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
3
Reference Systems and Planetology, Royal Observatory of Belgium, Brussels, Belgium
4
Dept. of Astrophysics, IMAPP, Radboud University Nijmegen, 6500 GL, Nijmegen, The Netherlands
5
Max Planck Institute for Astronomy, Koenigstuhl 17, 69117 Heidelberg, Germany
arXiv:2108.02907v1 [astro-ph.SR] 6 Aug 2021
Received June 17, 2021 / Accepted August 5, 2021
ABSTRACT
Context. Slowly pulsating B (SPB) stars are main-sequence multi-periodic oscillators that display non-radial gravity modes. For a
fraction of these pulsators, 4-year photometric light curves obtained with the Kepler space telescope reveal period spacing patterns
from which their internal rotation and mixing can be inferred. In this inference, any direct resonant mode coupling has usually been
ignored so far.
Aims. We re-analysed the light curves of a sample of 38 known Kepler SPB stars. For 26 of those, the internal structure, including
rotation and mixing, was recently inferred from their dipole prograde oscillation modes. Our aim is to detect direct nonlinear resonant
mode coupling among the largest-amplitude gravity modes.
Methods. We extract up to 200 periodic signals per star with five different iterative prewhitening strategies based on linear and
nonlinear regression applied to the light curves. We then identify candidate coupled gravity modes by verifying whether they fulfil
resonant phase relations.
Results. For 32 of 38 SPB stars we find at least 1 candidate resonance that is detected in both the linear and the best nonlinear
regression model fit to the light curve and involves at least one of the two largest-amplitude modes.
Conclusions. The majority of the Kepler SPB stars reveal direct nonlinear resonances based on the largest-amplitude modes. These
stars are thus prime targets for nonlinear asteroseismic modelling of intermediate-mass dwarfs to assess the importance of mode
couplings in probing their internal physics.
Key words. asteroseismology – stars: oscillations (including pulsations) – stars: early-type – stars: variables: general – stars: rotation
– methods: data analysis
1. Introduction De Cat & Aerts 2002; De Cat et al. 2005, 2007), but their aster-
oseismic potential only became clear when CoRoT (Auvergne
Three decades ago Waelkens (1991) first coined the group of et al. 2009) and Kepler (Borucki et al. 2010) space photometry
“slowly pulsating B” (SPB) stars to describe mid-B variable stars became available (see e.g., Degroote et al. 2010; Balona et al.
that display low-frequency oscillations with physical properties 2011, 2015; Pápics et al. 2012, 2014, 2015, 2017; Szewczuk &
similar to 53 Persei variables (see e.g., Waelkens & Rufener Daszyńska-Daszkiewicz 2018; Pedersen et al. 2021; Szewczuk
1985, and references therein). The SPB stars are mid-to-late B et al. 2021, for recent examples of light curves). Because of their
main-sequence (MS) stars with masses ranging from ∼ 3 to ∼ wealth in number of excited and identified oscillation modes,
9 M (De Cat & Aerts 2002; Aerts et al. 2010; Pedersen et al. they have been the target of various asteroseismic modelling ef-
2021) displaying all levels of rotation, from very slow up to the forts, which focused on inferring the internal mixing and rota-
critical rotation rate (Pedersen et al. 2021). A distinct feature tion (e.g., Moravveji et al. 2015, 2016; Triana et al. 2015; Pápics
is their multi-periodic photometric and spectroscopic variabil- et al. 2017; Pedersen et al. 2021; Michielsen et al. 2021) and
ity, with periods typically ranging from ∼ 0.5 to ∼ 5.0 d and on mode excitation (e.g., Szewczuk & Daszyńska-Daszkiewicz
amplitudes typically less than ∼ 10 mmag (e.g., Aerts et al. 2018; Szewczuk et al. 2021). There have also been efforts to
2010; Pápics et al. 2017; Pedersen et al. 2021). Their variability characterise the magnetic field of a small number of SPB stars
is attributed to low-degree, high-radial order gravity (g) modes, (e.g. HD 43317) by analysing the frequency shifts of its g-modes
which are excited by the heat engine (i.e. κ) mechanism driven (Buysschaert et al. 2018; Prat et al. 2019, 2020). The influence
by an opacity enhancement from the iron-group elements, also of rotation on the g modes in SPB stars is more important than
called the Z Bump, at a temperature of ∼ 200 000 K (Gautschy the magnetic shifts and imply that almost all modes are in the
& Saio 1993; Dziembowski et al. 1993; Pamyatnykh 1999). gravito-inertial regime (Aerts et al. 2017, 2019). The effects of
The oscillations in SPB stars are fairly well-characterised rotation, therefore, cannot be incorporated from a perturbative
from ground-based photometric and spectroscopic observations treatment of the Coriolis acceleration, which led many authors to
(e.g., Aerts et al. 1999; De Cat et al. 2000; Mathias et al. 2001; adopt the so-called traditional approximation of rotation (TAR;
Article number, page 1 of 16A&A proofs: manuscript no. main
see e.g., Longuet-Higgins 1968; Lee & Saio 1997; Townsend onance condition exactly, which can distort the period spacing
2003; Degroote et al. 2009b; Mathis 2013) for pulsation compu- patterns relied upon in linear asteroseismology.
tations (e.g., Szewczuk & Daszyńska-Daszkiewicz 2018; Peder- The theoretical framework of resonant mode coupling, the
sen et al. 2021; Szewczuk et al. 2021; Michielsen et al. 2021). amplitude equation (AE) formalism, is described in numerous
Rapid rotation also deforms a star, causing it to become publications (e.g., Dziembowski 1982; Buchler & Goupil 1984;
oblate. The rotation rates of SPB stars cover the cases from Dziembowski & Krolikowska 1985; Buchler 1993; Buchler et al.
slight to major deformation. However, the detected and iden- 1997; Van Hoolst & Smeyers 1993; Van Hoolst 1994a,b, 1995).
tified modes have their dominant mode energy deep inside the It has the inherent validity assumption that the (linear) growth
star, close to the convective core, where spherical symmetry still rates, γ, of the modes are small compared to their angular fre-
applies well. This justifies a treatment of g modes in the TAR quencies, which is generally assumed to be satisfied for observed
(Mathis & Prat 2019; Henneco et al. 2021; Dhouib et al. 2021). MS oscillators (see e.g., Barceló Forteza et al. 2015; Bowman
An important consequence of the rapid rotation of some SPB et al. 2016). The AEs constitute a set of coupled first-order com-
stars is the occurrence of mode coupling between overstable con- plex nonlinear ordinary differential equations and predict the
vective modes based on frozen-in core convection and envelope temporal behaviour of the mode amplitudes and phases. Three
g modes excited by the κ mechanism (Lee 2021). main regimes of mode interaction are distinguished (as explained
SPB stars have so far been modelled from linear pulsation in Buchler et al. 1997):
theory. Period spacing patterns constructed from the frequencies
of identified g modes of consecutive radial order and the same 1. If two or more modes are close to being in resonance, such
angular degree and azimuthal order, are the most important tool that Dg ≡ δω/γ is of order 1 or smaller (where δω is the
used in linear asteroseismic modelling. Such patterns can display angular frequency mismatch or angular frequency detuning),
oscillatory features attributable to chemical gradients inside the and a stable fixed point (FP) solution exists for the AEs, for
star (e.g., Miglio et al. 2008). Moreover, they display a slope due which the mode amplitudes and phases are constant over
to the rotation rate near the convective core (e.g., Bouabid et al. time, the frequencies are locked. These locked frequencies
2013; Pápics et al. 2017). Some SPB stars, however, can show can be substantially different from their non-resonant coun-
structure in their g-mode period spacing patterns that is hard to terparts.
explain without incorporating additional physical effects in the
oscillation model, such as rotational deformation (e.g., Dhouib 2. With an increasing angular frequency mismatch, δω, the FP
et al. 2021) or (internal) magnetism (see e.g., Van Beeck et al. solution becomes unstable or disappears, and a bifurcation
2020), or resonances of envelope g modes with inertial modes in to another solution takes place, with amplitude (and phase)
the core (e.g., Ouazzani et al. 2020; Saio et al. 2021). modulation. The larger the value of Dg , the smaller the am-
plitude of the modulation. This is called the ‘intermediate’
The input to asteroseismic modelling based on linear oscil-
regime (Goupil et al. 1998).
lation theory are the independent frequencies extracted from an
observed time series (e.g., a light curve), where frequency ex- 3. At large δω one encounters the solution where frequencies
traction is usually done using an iterative prewhitening proce- are approximately equal to their (unlocked) non-resonant
dure (e.g., Degroote et al. 2009b). However, non-sinusoidal light values. The nonlinear nonresonant frequencies have a small
curves give rise to harmonics and combination frequencies (de- frequency shift compared to their linear counterparts: this is
pendent frequencies) in the amplitude spectrum (Pápics 2012; the mildly nonlinear regime in which linear asteroseismol-
Kurtz et al. 2015). These combination frequencies are therefore ogy operates, also called the ‘nonresonant’ regime (Goupil
also included in the extracted frequencies. The phases and ampli- et al. 1998).
tudes of the independent frequencies, as well as the frequencies,
amplitudes, and phases of the combination frequencies are sub- In addition, a narrow hysteresis (transitory) regime exists in
sequently neglected during linear asteroseismic modelling. On between the frequency-locked and ‘intermediate’ regime, in
the other hand, frequency perturbations within g-mode period which amplitudes are modulated, but frequencies (or equiva-
spacing patterns are expected if nonlinear mode coupling is ac- lently, phases) remain constant (Buchler et al. 1997).
counted for (e.g., Buchler & Goupil 1984; Van Hoolst 1996). The study of the relationships among amplitudes and phases,
Hence, this may be an alternative explanation for the deviations which provide the observables to be matched to the theoretical
in the period spacing patterns of SPB stars. Such frequency shifts predictions from the AEs, is a standard approach for the analy-
may be small if the g modes are only moderately nonlinear, as sis of light curves of RR Lyr stars and Cepheids (as pioneered
is the case for most ‘well-observed’ nonradial pulsators such as by Simon & Lee 1981). It is also a well-established technique
white dwarfs or δ Sc stars (Buchler et al. 1997), yet this remains for unravelling nonlinearities in the light curves of oscillating
to be verified for SPB stars. white dwarf stars (e.g., Wu 2001; Zong et al. 2016b), and hot B
As detailed in Buchler et al. (1997), resonances among (ex- subdwarf stars (e.g., Zong et al. 2016a). Yet, it has not received
cited) non-radial oscillation modes also play an important role much attention for oscillating MS stars (Lee 2012; Bowman &
in mode selection and interaction, and can lead to additional Kurtz 2014; Kurtz et al. 2015; Bowman et al. 2016, 2021). In
constraints on mode identification. Resonant interactions among this work, we provide the identification of candidate direct res-
modes can further hamper pattern identification due to so-called onances among the extracted oscillation frequencies in the light
frequency or phase locking. ToPcomplicate pattern identification curves of 38 SPB stars, and test the impact of different iterative
the exact resonance relation, i ni ωi = 0, need not be satis- prewhitening strategies on the results. In Sect. 2 we summarise
fied for linear mode frequencies. It suffices that linear mode fre- the characteristics of our sample, the strategies we employ to
quencies are in near resonance, such that the exact resonance analyse the light curves, and discuss how we search for candi-
relation between angular frequencies ω j is satisfied aside from date direct resonances. We discuss the light curve analysis strat-
a small angular frequency difference δω. Nonlinear frequency egy selection process in Sect. 3, and the results of the candidate
shifts can then lead to a nonlinear frequency locking or phase resonance search in Sect. 4. A summary of our conclusions and
locking, where the nonlinear locked frequencies satisfy the res- future prospects can be found in Sect. 5.
Article number, page 2 of 16J. Van Beeck et al.: Nonlinear g-mode resonances in SPB stars
Table 1. Five iterative prewhitening strategies employed in this work. step (Bowman et al. 2016). It requires each signal to have an am-
plitude S/N ≥ 4 (Breger et al. 1993) computed in a window of
Strategy Hinting Stop criterion Final optimisation a 1 d−1 around the target frequency, and fits a model comprising a
sum of sinusoids, F(ti ), to the light curve:
1 S/N S/N < 4.0 nonlinear
nf
2 A pA b > 0.05 nonlinear X h i
F(ti ) = β0 + A j sin 2πν j ti + φ j , (1)
3 A pLRT c > 0.05 nonlinear
j=1
4 S/N pLRT c > 0.05 nonlinear
5d A pLRT c > 0.05 linear for each ti , i = 1, . . . , nt , with nt the total number of timestamps
of the light curve. In this expression, n f is the number of fit-
Notes. Parameter hinting is either performed by selecting the highest- ted frequencies,
β0 is the y-intercept, A j denotes the amplitude,
amplitude peak (A) in the residual LS periodogram, or the peak with the ν j ≡ ω j /2π is the temporal frequency, and φ j is the phase of
highest amplitude S/N value using a 1 d−1 window (S/N). The selected fitted sinusoid j. The other four prewhitening strategies differ
peak is not closer than 2.5 / T to earlier extracted frequencies, where from this first one in the way they determine the initial parame-
T is the total time span of the data (Loumos & Deeming 1978). Lin-
ear interpolation between the over-sampled frequency grid points of the
ters for the optimisation (i.e. parameter hinting; see the notes of
Lomb-Scargle periodogram (Lomb 1976; Scargle 1982) is employed to Table 1), stop criterion, and whether a linear or nonlinear (multi-
determine the S/N value associated with the optimised frequency. parameter) least-squares optimisation is performed at the end of
(a)
Type of least-squares optimisation of the entire model at the end of each iterative prewhitening step. These differences are listed in
each iterative prewhitening step. (b) The p-value for a Z-test based on Table 1.
extracted amplitude. (c) The p-value for a likelihood ratio test (LRT) As is commonly done, we calculate the uncertainties for the
based on Bayesian information criterion (BIC) values for the nested extracted parameters, under the assumption that they are uncor-
light curve regression models. (d) The frequency extracted during the it- related, using
erative prewhitening step is optimised by selecting the largest-amplitude
signal in an over-sampled LS periodogram in a 1 d−1 region around the √
6 σnt 2 σ nt
r r
2
frequency hint. σ̂ν = D √ , σ̂A = D σnt , σ̂φ = D , (2)
π nt  T nt nt Â
2. Light curves and their analysis where σnt is the standard deviation of the residual signal (after
the regression model has been subtracted from the light curve),
2.1. The Pedersen (2020) SPB sample  is the optimised amplitude, T is the total time span of the data,
and where D is a correction factor for the correlated nature of
Our sample consists of 38 stars originally identified as SPB stars the light curve data, which can be estimated as the square root
based on Kepler photometry (Balona et al. 2011, 2015; McNa- of the average number of consecutive data points of the same
mara et al. 2012; Pápics et al. 2013, 2014, 2015, 2017; Zhang sign in the residual light curve (e.g., Schwarzenberg-Czerny
et al. 2018; Szewczuk & Daszyńska-Daszkiewicz 2018). We use 1991; Montgomery & O’Donoghue 1999; Schwarzenberg-
the light curves extracted from the 30-min cadence target pixel Czerny 2003; Degroote et al. 2009b). During each prewhitening
data by Pedersen (2020), who used customised pixel masks (see step we compute the Lomb-Scargle (LS) periodogram (Lomb
Pápics et al. 2017 for further details). After light curve extrac- 1976; Scargle 1982) of the (residual) light curve, with frequency
tion, Pedersen (2020) manually removed outliers, and separately grid step equal to 0.1Rν (where Rν ≡ T1 is the Rayleigh limit).
corrected each quarter for instrumental trends by fitting a low or- The start and end values of this grid are equal to 1.5 Rν and the
der polynomial to the data. Pedersen (2020) then converted the Nyquist frequency νNyq (νNyq = 1/(2 ∆t), ˜ where ∆t ˜ is the median
light curves to have flux units of ppm, and normalised them to cadence of the time series), respectively. The LS periodogram,
have an average flux of zero. together with a specific parameter hinting rule listed in Table
Contrary to previous analyses of Kepler SPB stars by 1, are used to provide the preliminary amplitude and frequency
Szewczuk & Daszyńska-Daszkiewicz (2018), Pedersen (2020), hint, taking into account the frequency resolution (Loumos &
Szewczuk et al. (2021) and Pedersen et al. (2021), we use five Deeming 1978), hereafter referred to as the LD78 resolution.
different iterative prewhitening strategies, each of which produce The preliminary phase hint is subsequently obtained from a lin-
a final regression model fit for the light curve. Our approach is ear least-squares fit to the residual light curve, where frequencies
motivated by the fact that iterative prewhitening techniques vary and amplitudes in Eqn. (1) are fixed at either their frequency
in the literature, while it is not assessed how these various ap- and amplitude hint values (the frequency being extracted), or
proaches influence frequency extraction results. Our compara- the values they had in the previous prewhitening step (all other
tive capacity assessment of the different iterative prewhitening frequencies). The phase hinting optimisation employs the Trust-
procedures is an important aspect of the hunt for nonlinear reso- region reflective least-squares regression method implemented
nances in SPB stars. in the Python package lmfit (Newville et al. 2020), whose ex-
act configuration is specified in Appendix A. For strategies 1
2.2. SPB light curve analysis to 4 the optimised parameter hints are then fed to the nonlin-
ear least-squares optimiser at the end of this prewhitening step
We employ five prewhitening strategies to derive frequencies, to optimise the frequencies, amplitudes and phases of the entire
amplitudes and phases from the light curves, which emulate var- model. For strategy 5, however, these parameter hints are only
ious approaches taken in the literature. The first timestamp of used in a linear least-squares optimisation of the amplitudes and
each light curve is subtracted from all timestamps of that light phases, where the frequencies are fixed at the values determined
curve to provide a consistent zero point in time for phase calcu- from an over-sampled LS periodogram in a 1 d−1 region around
lation. Strategy 1 uses a nonlinear least-squares fit of the entire the preliminary frequency hint (signal extracted during this iter-
multi-parameter model in the time domain at each prewhitening ative prewhitening step), and the values they had in the previous
Article number, page 3 of 16A&A proofs: manuscript no. main
iteration step (other signals). The Levenberg-Marquardt method where φdaughter is the extracted phase of the daughter frequency,
implemented in lmfit (Newville et al. 2020) is used for this opti- and φparent, i is the extracted phase of its i-th parent frequency. If
misation, whose exact configuration is specified in Appendix A. for a given candidate combination of parent(s) and daughter fre-
Each iterative prewhitening process continues until 200 fre- quencies Eqns (4) and (5) are satisfied, this pair is classified as a
quencies have been extracted in this way, or until the speci- candidate resonance. We define P the combination order o of this
fied stop criterion as in Table 1 is reached. After the frequency candidate resonance as: o = i |ni |. Following Vuille & Bras-
extraction, we verify that the amplitude of each extracted fre- sard (2000) we compute the relative amplitude of the candidate
quency is significant at the 95% level using a Z-test, whilst tak- resonance, Ar , in the following way:
ing the LD78 resolution into account. If at least one insignifi-
cant extracted frequency is detected, the last extracted insignifi- Adaughter
Ar = Y |ni | . (6)
cant frequency is removed. After that frequency is removed, the Aparent, i
entire model is re-optimised, and a new significance check is i
performed. This iterative process, which we refer to as the fre-
quency filtering step, continues until no further insignificant fre- In practice, we limit ourselves to identifying up to second-
quencies occur in the regression model. order combinations of parent frequencies (o = 2). For each
The regression models for each of the prewhitening strate- potential candidate direct resonance, we then verify whether
gies that have undergone frequency filtering are then compared Eqn. (4) is satisfied to within 1 Rayleigh limit Rν at the 1-σ
with one another. To quantify the quality of the regression model, level:
we rely on the residual sum of squares, weighted by the total
variance in the observed light curve and by a factor including ∃νdaughter : νdaughter − ν − σνdaughter − σν ≤ Rν , (7)
the total number of free parameters of the regression model, n p ,
where ν ≡ i ni νparent, i is a combination frequency composed
P
versus nt . To end up with a maximum likelihood estimator, we
subsequently use the scaled fraction of variance, fsv , which we of other extracted frequencies, with propagated uncertainty σν .
define as The comparison with Rν hence measures whether ν is resolved
Pnt with respect to νdaughter . We use this practical approximation for
i (yi − F(ti ))2 nt − 1
( ) assigning candidate resonance identifiers, to avoid use of the
fsv ≡ 1 − P , (3) theoretical criterion by Buchler et al. (1997), |δω|/γ . O(1),
i (yi − ȳ)
nt 2 nt − n p
setting the approximate limits of parameter space for which an
FP solution with frequency locking exists, as it relies on model-
Pnt yi denotes the time series signal at timestamp ti , and ȳ =
where
dependent nonadiabatic oscillation calculations for γ, while we
i yi /nt is the mean signal of the time series. The higher the
quality of the regression model, the closer fsv approaches unity want to establish a data-driven approach. If Eqn. (7) holds, we
while we punish for a higher number of free parameters. further verify whether the relative phase Φ of the candidate com-
bination satisfies Eqn. (5) to within 1 σ:
π
2.3. Candidate resonant Kepler SPB oscillations ∃k ∈ [−4, −3, . . . , 4] : Φ − σΦ ≤ k · ≤ Φ + σΦ . (8)
2
In our identification of direct resonant mode interactions we fo-
cus on FP solutions of the AEs, for which frequency locking
persists (Buchler et al. 1997). Any such solutions have parent- 3. Results: light curve regression model capacity
daughter frequency combinations that approximately satisfy the In this section the resulting regression models obtained for each
(direct) resonance relation. Our aim is to extract information of the five prewhitening strategies applied to all 38 stars in our
on the least complicated mode interactions, hence, we use strict sample are compared. The metrics used for this comparison are
candidate resonance identification criteria. The first criterion re- the fsv value defined in Eqn. (3), and the (residual) light curves
quires that a frequency combination of a candidate resonance and their LS periodogram. The fsv metric measures how well
satisfies the following relation: the five regression models describe the light curve of any given
X star, where we recall that Eqn. (1) assumes that each oscillation
∃νdaughter : δν ≡ νdaughter − ni νparent, i ≈ 0 d−1 , (4) mode has constant amplitude, frequency, and phase. Under this
i assumption and using the criteria in Table 1, we define the best
where νdaughter is the daughter frequency, νparent, i is its i-th parent regression model as the one with the highest fsv . Any violation
frequency, ni ∈ Z, and where δν is the frequency detuning, for of the assumed constant amplitudes, phases, or frequencies re-
which |δν| < νdaughter holds. This follows the parent-daughter ter- sults in lower values of fsv along with the occurrence of more
minology introduced in Degroote et al. (2009b). To further char- harmonics and combination frequencies.
acterise and verify the direct candidate resonance, and follow- Based on the fsv metric, we divide the stars in our sample into
ing Buchler et al. (1997), Vuille (2000) and Vuille & Brassard two pseudoclasses: (i) stars for which the light curve regression
(2000), we compute relative phases and amplitudes. The second models attain fsv values above 0.9 after frequency filtering for all
candidate resonance identification criterion then consists of veri- five iterative prewhitening strategies; and (ii) stars for which at
fying whether the corresponding relative phases Φ of the combi- least one of these models attains a value of fsv < 0.9. For brevity,
nations that satisfy Eqn. (4) also satisfy the following (resonant) we instead refer to strategies ‘having’ fsv values above 0.9 in the
relative phase relation (see e.g., Fig. 13 in Van Hoolst 1995, in remainder of the text. This classification accentuates the appar-
which Γ 0 and σ denote the FP relative phase and frequency de- ent bimodality in the distribution of the fsv metric values that can
tuning, respectively, as well as Fig. 1 in Buchler et al. 1997): be seen in Fig. 1. Half of the 38 SPB stars in our sample have
fsv ≥ 0.9 for all five prewhitening strategies and thus belong to
pseudoclass 1, hereafter referred to as the ‘high- fsv ’ stars. We
X π discuss their light curve regression model selection process in
∃k ∈ [−4, −3, . . . , 4] : Φ ≡ φdaughter − ni φparent, i ≈ k· , (5)
i
2 subsection 3.1. We additionally subdivide the stars belonging the
Article number, page 4 of 16J. Van Beeck et al.: Nonlinear g-mode resonances in SPB stars
second pseudoclass (the ‘low- fsv ’ stars) into two additional pseu- of the periodic signals in the light curves. For KIC005941844,
doclasses based on whether outburst-like features are detected in the iteration progress curves indicate that for all five iterative
the light curves, and thus end up with three pseudoclasses that prewhitening strategies, the optimal or near-optimal model has
describe the whole sample. The light curve regression model se- been obtained, before any additional post-processing frequency
lection process for these two additional pseudoclasses, the high filtering. The frequency filtering step removes unresolved fre-
mode density starands and the outbursting stars, are discussed in quencies and frequencies that do not have significant amplitudes
subsection 3.2, both guided by particular example stars. The per- (at a 95% confidence level), so that the final light curve regres-
centages of explained scaled variance fsv averaged over all stars sion models are obtained. The numbers of frequencies retained
in our sample, and averaged over the stars belonging to each of in the regression models after this frequency filtering step (i.e.
these pseudoclasses are shown in Fig. 2. According to this ‘sam- the numbers of final resolved significant extracted frequencies)
ple’ metric, high- fsv stars are explained well by the regression are indicated in Fig. 1 by the ne parameter for each strategy ap-
models belonging to the five prewhitening strategies. However, plied to the light curve of each star.
significantly differing values of fsv can be noted for low- fsv stars. The differences among the fsv values of the regression mod-
For the sample of 38 SPB stars as a whole, the overall highest- fsv els for all five strategies are small for the high- fsv stars, as can be
regression model is the one delivered by prewhitening strategy noticed when comparing these values listed for KIC005941844
3, as will be discussed further below. The best regression model in Fig. 1. Yet, the numbers of extracted frequencies, ne , differ
and other properties for all individual stars in our sample are dis- strongly. Good correspondence with respect to the number of
cussed in Appendix B. extracted frequencies for strategies 2 and 3 occurs for almost
all high- fsv stars. For strategies 1 and 4, the parameter hinting
selects the signal in the residual LS periodogram with the high-
3.1. High- fsv stars
est S/N value. Because we analyse g-mode oscillators, the pa-
We discuss the light curve regression model selection process for rameter hinting for these strategies renders them prone to select
high- fsv stars guided by the example of KIC005941844, classi- a new frequency in a mode-scarce region of the periodogram,
fied as an SPB star by McNamara et al. (2012) and analysed by which is illustrated in Fig. 4, with the mode-scarce regions typ-
Pedersen (2020) and Pedersen et al. (2021). Its light curve is well ically found at higher frequencies. Signals in mode-scarce re-
explained by sinusoids of constant amplitude and frequency, as gions of a typical LS periodogram of a g-mode oscillator are of
demonstrated by its high fsv values and the light curve in the top lower amplitude than those in mode-dense regions. These sig-
panel of Fig. 3. The light curves of high- fsv stars are, in gen- nals therefore contribute less to fsv , explaining why for many
eral, (nearly) sinusoidal, because a large degree of variance is high- fsv stars, including KIC005941844, the stop criterion is
explained by the model generated by Eqn. (1). The most promi- triggered earlier for strategy 4 than for the other strategies. Con-
nent data gaps in the light curve of KIC005941844 are caused by versely, the parameter hinting for strategies 2, 3, and 5 selects
the loss of Kepler’s CCD module 3, one year into the mission. the largest-amplitude signal, and thus targets the mode-dense re-
The middle panel of Fig. 3 shows a comparison of the values gions of such a typical periodogram. This hence reiterates the
of the metrics important for the stop criteria of all five strate- important role that mode density plays in frequency extraction,
gies, throughout the iterative prewhitening process. The values even for Kepler light curves, which offer the most optimal fre-
of these metrics as a function of the prewhitening step number quency resolution for space photometry available today. More-
indicate the progress of that process before frequency filtering over, because different strategies focus their extraction efforts in
is carried out, and are henceforth referred to as the ‘iteration different frequency regimes of the periodogram, the physical in-
progress curves’. If the gradient of the Bayesian information cri- formation extracted from these differing light curve regression
terion (BIC) iteration progress curve (strategy 3, 4, or 5) con- models is different and can be complementary.
verges to a value of zero, the regression model is approaching The effect of a high mode density in the periodogram of
the optimal model according to the likelihood ratio test (LRT). KIC005941844 is clearly visible in the bottom panel of Fig. 3:
The LRT compares the nested light curve regression models of dozens of small-amplitude signals persist in the residual LS pe-
this and the previous step based on the likelihoods inferred from riodogram, which are unresolved with respect to the LD78 cri-
the BIC values. A significant upward trend in the BIC iteration terion. Such unresolved signals can cause modulation of am-
progress curve increases the p-value of the LRT to a value greater plitudes and frequencies because they can interfere with other
than the stop criterion, with the significance of that trend depend- modes (e.g., Bowman et al. 2016) or be residual artefacts from
ing on the difference in BIC values and number of optimisation imperfect prewhitening under the assumption that all oscillations
parameters of the nested regression models. The S/N iteration have constant frequencies, amplitudes and phases. In this case
progress curve (strategy 1) represents the signal-to-noise ratio of the amplitudes of the extracted frequencies are large compared
the frequency extracted during each iterative prewhitening step. to the amplitudes of these unresolved signals, hence fsv values
If it falls below the pre-set limit S /N = 4.0 (Breger et al. 1993), are not strongly affected by such effects and remain high.
the stop criterion is triggered and iterative prewhitening stops. A sometimes neglected point in frequency analysis is that
Finally, the A/σ̂A iteration progress curve represents the ampli- some of the free parameters in the fit to the light curve covary
tude of the frequency extracted during each iterative prewhiten- with other free parameters in the fit. Particularly strong covari-
ing step, in terms of its standard deviation. Iterative prewhitening ances exist between the extracted frequencies and their corre-
for strategy 2 is stopped if the amplitude of the frequency ex- sponding phases. As an example, neglecting such frequency-
tracted is consistent with 0.0 ppm at 95% confidence level (i.e. phase covariances leads to differences in the frequency val-
A/σ̂A ≤ 1.96). ues of the two largest-amplitude extracted frequencies of
The iteration progress curves provide a metric that dictates KIC005941844 at 2-σ level when comparing their amplitudes
whether a light curve regression model is nearing its optimal pa- for the same mode frequencies extracted with strategy 5. Covari-
rameters. This needs to be investigated on a star-to-star basis, be- ances among regression parameters not only influence the esti-
cause it intrinsically depends on the oscillation mode density in mation of relative phases that are crucial to identifying candidate
the (residual) LS periodograms and amplitudes attained by any resonances, but can also exert influence on asteroseismic mod-
Article number, page 5 of 16A&A proofs: manuscript no. main
fsv
003240411 fsv: 0.6832 | ne: 70 fsv: 0.7225 | ne: 175 fsv: 0.7229 | ne: 177 fsv: 0.5865 | ne: 17 fsv: 0.6367 | ne: 28
003459297 fsv: 0.9918 | ne: 138 fsv: 0.9933 | ne: 180 fsv: 0.9933 | ne: 179 fsv: 0.9901 | ne: 98 fsv: 0.9913 | ne: 94
003756031 fsv: 0.8709 | ne: 100 fsv: 0.9146 | ne: 181 fsv: 0.9146 | ne: 181 fsv: 0.5809 | ne: 13 fsv: 0.9054 | ne: 152
003839930 fsv: 0.9898 | ne: 89 fsv: 0.9928 | ne: 189 fsv: 0.9928 | ne: 189 fsv: 0.9818 | ne: 40 fsv: 0.9902 | ne: 91
003865742 fsv: 0.9627 | ne: 81 fsv: 0.9777 | ne: 190 fsv: 0.9776 | ne: 189 fsv: 0.9534 | ne: 46 fsv: 0.9767 | ne: 188
004930889 fsv: 0.9865 | ne: 84 fsv: 0.9900 | ne: 174 fsv: 0.9900 | ne: 174 fsv: 0.9684 | ne: 16 fsv: 0.9887 | ne: 129 0.9
004936089 fsv: 0.9960 | ne: 83 fsv: 0.9977 | ne: 180 fsv: 0.9977 | ne: 180 fsv: 0.9940 | ne: 48 fsv: 0.9971 | ne: 124
004939281 fsv: 0.7094 | ne: 60 fsv: 0.7562 | ne: 141 fsv: 0.7571 | ne: 143 fsv: 0.6085 | ne: 8 fsv: 0.7463 | ne: 95
005084439 fsv: 0.9749 | ne: 114 fsv: 0.9801 | ne: 182 fsv: 0.9801 | ne: 182 fsv: 0.9785 | ne: 160 fsv: 0.9755 | ne: 86
005309849 fsv: 0.9831 | ne: 146 fsv: 0.9877 | ne: 188 fsv: 0.9878 | ne: 189 fsv: 0.9749 | ne: 93 fsv: 0.9876 | ne: 189
005941844 fsv: 0.9969 | ne: 119 fsv: 0.9975 | ne: 183 fsv: 0.9975 | ne: 183 fsv: 0.9915 | ne: 28 fsv: 0.9973 | ne: 184 0.8
006352430 fsv: 0.9917 | ne: 96 fsv: 0.9933 | ne: 190 fsv: 0.9933 | ne: 190 fsv: 0.9870 | ne: 39 fsv: 0.9932 | ne: 187
006462033 fsv: 0.9169 | ne: 153 fsv: 0.9388 | ne: 190 fsv: 0.9388 | ne: 190 fsv: 0.8636 | ne: 74 fsv: 0.9129 | ne: 95
006780397 fsv: 0.9683 | ne: 104 fsv: 0.9741 | ne: 180 fsv: 0.9741 | ne: 180 fsv: 0.9568 | ne: 53 fsv: 0.9737 | ne: 173
007630417 fsv: 0.6402 | ne: 144 fsv: 0.6857 | ne: 179 fsv: 0.6853 | ne: 178 fsv: 0.6292 | ne: 127 fsv: 0.6782 | ne: 155
007760680 fsv: 0.9956 | ne: 84 fsv: 0.9954 | ne: 64 fsv: 0.9954 | ne: 64 fsv: 0.9937 | ne: 27 fsv: 0.9956 | ne: 103
008057661 fsv: 0.7614 | ne: 145 fsv: 0.8119 | ne: 190 fsv: 0.8119 | ne: 190 fsv: 0.7214 | ne: 101 fsv: 0.8066 | ne: 176 0.7
008087269 fsv: 0.8839 | ne: 116 fsv: 0.8902 | ne: 160 fsv: 0.8905 | ne: 161 fsv: 0.8494 | ne: 44 fsv: 0.8887 | ne: 140
008255796 fsv: 0.8336 | ne: 72 fsv: 0.8760 | ne: 185 fsv: 0.8760 | ne: 185 fsv: 0.8072 | ne: 44 fsv: 0.8688 | ne: 149
KIC
008324482 fsv: 0.9408 | ne: 133 fsv: 0.9502 | ne: 191 fsv: 0.9502 | ne: 191 fsv: 0.9188 | ne: 65 fsv: 0.9415 | ne: 127
008381949 fsv: 0.4390 | ne: 55 fsv: 0.5992 | ne: 182 fsv: 0.5992 | ne: 182 fsv: 0.3004 | ne: 23 fsv: 0.5489 | ne: 111
008459899 fsv: 0.7283 | ne: 71 fsv: 0.8106 | ne: 177 fsv: 0.8106 | ne: 177 fsv: 0.5713 | ne: 15 fsv: 0.7415 | ne: 68 0.6
008714886 fsv: 0.9748 | ne: 110 fsv: 0.9813 | ne: 188 fsv: 0.9813 | ne: 188 fsv: 0.9531 | ne: 52 fsv: 0.9809 | ne: 187
008766405 fsv: 0.9813 | ne: 100 fsv: 0.9824 | ne: 158 fsv: 0.9824 | ne: 159 fsv: 0.9761 | ne: 40 fsv: 0.9822 | ne: 155
009020774 fsv: 0.9893 | ne: 108 fsv: 0.9904 | ne: 191 fsv: 0.9903 | ne: 189 fsv: 0.9895 | ne: 127 fsv: 0.9890 | ne: 63
009227988 fsv: 0.9184 | ne: 57 fsv: 0.9504 | ne: 173 fsv: 0.9504 | ne: 173 fsv: 0.9157 | ne: 54 fsv: 0.9495 | ne: 158
009715425 fsv: 0.4454 | ne: 45 fsv: 0.4411 | ne: 53 fsv: 0.4439 | ne: 54 fsv: 0.3357 | ne: 24 fsv: 0.4341 | ne: 57
009964614 fsv: 0.7276 | ne: 89 fsv: 0.8145 | ne: 186 fsv: 0.8145 | ne: 186 fsv: 0.5846 | ne: 23 fsv: 0.8074 | ne: 185 0.5
010220209 fsv: 0.9479 | ne: 106 fsv: 0.9533 | ne: 185 fsv: 0.9533 | ne: 185 fsv: 0.9514 | ne: 192 fsv: 0.9399 | ne: 41
010285114 fsv: 0.7015 | ne: 130 fsv: 0.7133 | ne: 163 fsv: 0.7138 | ne: 164 fsv: 0.6343 | ne: 35 fsv: 0.7018 | ne: 101
010526294 fsv: 0.9701 | ne: 68 fsv: 0.9706 | ne: 76 fsv: 0.9707 | ne: 77 fsv: 0.9655 | ne: 39 fsv: 0.9695 | ne: 76
010536147 fsv: 0.7284 | ne: 101 fsv: 0.7921 | ne: 189 fsv: 0.7921 | ne: 189 fsv: 0.6629 | ne: 31 fsv: 0.7921 | ne: 189
010658302 fsv: 0.9338 | ne: 97 fsv: 0.9378 | ne: 116 fsv: 0.9419 | ne: 171 fsv: 0.9170 | ne: 51 fsv: 0.9310 | ne: 80 0.4
011152422 fsv: 0.3526 | ne: 36 fsv: 0.4921 | ne: 161 fsv: 0.4934 | ne: 166 fsv: 0.3985 | ne: 65 fsv: 0.4757 | ne: 115
011360704 fsv: 0.4295 | ne: 53 fsv: 0.5201 | ne: 175 fsv: 0.5201 | ne: 175 fsv: 0.3622 | ne: 19 fsv: 0.5060 | ne: 142
011454304 fsv: 0.7203 | ne: 101 fsv: 0.7538 | ne: 177 fsv: 0.7541 | ne: 178 fsv: 0.6027 | ne: 32 fsv: 0.7241 | ne: 88
011971405 fsv: 0.6779 | ne: 76 fsv: 0.7221 | ne: 175 fsv: 0.7218 | ne: 174 fsv: 0.6204 | ne: 25 fsv: 0.6712 | ne: 57
012258330 fsv: 0.8983 | ne: 133 fsv: 0.9077 | ne: 188 fsv: 0.9077 | ne: 188 fsv: 0.8808 | ne: 77 fsv: 0.9078 | ne: 188
1 2 3 4 5
Prewhitening strategy
Fig. 1. Comparison of the scaled fraction of variance fsv values of the resulting light curve regression models obtained for the different prewhitening
strategies applied to the stars in the sample, with colouring as a function of fsv . Each cell contains the fsv value as computed using Eqn. (3), as well
as the total number of frequencies extracted using this approach, ne . The number ne and the fsv value are computed after the frequency filtering step
has removed any unresolved frequencies with respect to the LD78 criterion and any signals that have insignificant amplitudes (at 95% confidence),
as discussed in section 2.2.
elling relying on the frequency values (Bowman & Michielsen, amplitudes, one should not use classical statistical model selec-
submitted). tion criteria to choose among the strategies 1 to 5. The case of
KIC005941844 illustrates this: its five fsv values differ by less
From a purely mathematical point of view, parsimony dic- than ∼ 6 · 10−3 but the different prewhitening strategies lead to
tates that any nested regression model that explains the same vastly different ne values. A classical model selection procedure
scaled fraction of variance as a more complex model (i.e. from via a BIC or LRT applied to the models based on the different
the same model strategy but containing more free parameters) strategies by just using their fsv alone would fail, as it would
should be preferred over that more complex model, because of favour the model obtained by strategy 4 for KIC005941844.
bias-variance trade-off (Claeskens & Hjort 2008). For this rea- However, it is clear that the other four regression models with
son, one can use information criteria for model selection when the vastly larger numbers of extracted modes capture more as-
the regression models are nested, as per each model strategy sep- trophysical information on, for example, the number of ex-
arately. However, one cannot use such criteria to distinguish be- cited modes, potential mode interactions or rotational multiplets.
tween the capacity of the different regression models based on Hence, the aim of finding a good regression model (high fsv )
different stopping criteria and optimisation methods. Moreover, along with a maximum number of resolved oscillation frequen-
the aim of any asteroseismic data analysis is to deduce as many cies of significant amplitude (high ne ) favour strategies 3, 4 or
significant and resolved oscillation frequencies as possible (i.e. 5 for this star. These three regression models preferentially se-
we wish to maximise ne as listed in Fig. 1). Since ne is a discrete lect signals at low frequencies, as shown in Fig. 4. The interplay
unknown that we want to maximise, within the adopted con- between the physical and (purely) mathematical considerations
ditions on frequency resolution and significance of the modes’
Article number, page 6 of 16J. Van Beeck et al.: Nonlinear g-mode resonances in SPB stars
all stars high mode density stars
high-fsv stars outbursting stars
100
Explained scaled variance (%)
80
60
97.90 %
97.87 %
97.63 %
97.34 %
96.62 %
86.23 %
86.21 %
85.01 %
83.27 %
80.90 %
80.91 %
78.84 %
78.61 %
75.63 %
40
66.47 %
60.84 %
60.75 %
58.92 %
55.27 %
20 49.33 %
×105
0 9.0 Prewhitening 70
1 2 3 4 5 strategy 40
Prewhitening strategy 1
2 60 35
8.5
3
Fig. 2. Total percentage of explained scaled variance fsv averaged over 4 50 30
all 38 stars in the sample (‘all stars’) for each of the five prewhitening 8.0 5 25
40
A
strategies. This averaged percentage is also displayed for each of the
S/N
BIC
A/
three pseudoclasses introduced in Section 3. 20
7.5 30
15
in selecting the optimal iterative prewhitening model is chal- 20
7.0 10
lenging. For the SPB stars with high fsv , Fig. 2 shows all five
prewhitening strategies to be almost equivalent by just looking 10
5
at the summed fsv for the 19 stars. However, adding the quest 6.5 S/N = 4.0
to maximise the number of resolved modes gives preference to 0 25 50 75 100 125 150 175 200
strategies 2 or 3. In any case, KIC005941844’s large-amplitude Prewhitening step
oscillations and small-amplitude signals unresolved with respect /2 ( Hz)
to the LD78 resolution, as well as its high fsv values, make it a 0 50 100 150 200 250
prime target for nonlinear asteroseismic analysis. 14000 /2 ( Hz)
5 10 15 20 25
12000 400
3.2. Low- fsv stars
10000 300
A (ppm)
A (ppm)
High mode densities are observed in the LS periodograms of all
low- fsv stars, because dozens of potential signals are left unre- 8000 200
solved with respect to the LD78 resolution. Six of the nineteen
100
low- fsv stars furthermore display outburst-like features: marked 6000
departures from the baseline brightness of the original light 0
curve, which are in phase with departures from the baseline in 4000 0.5 1.0 1.5 2.0 2.5
/2 (d 1)
the residual light curve. We make a specific distinction between
those stars that display these features in their light curves, and 2000
stars that do not. The former are referred to as ‘outbursting stars’,
0
whereas the latter are called ‘high mode density stars’. The dis- 0 5 10 15 20 25
cussion of the light curve regression model selection for low- fsv /2 (d 1)
stars is therefore guided by examples from both types of low-
fsv stars and the overall performance of the regression models in Fig. 3. Top: Full light curve of the high- fsv star KIC005941844 (left),
Fig. 2 is split up accordingly. and a small part of the light curve indicating its multi-periodic constant-
frequency nature (right). Middle: Comparison of the different stop cri-
Oscillation mode density is expected to increase with in- teria used for the five prewhitening strategies throughout the iterative
creasing rotation rate due to rotational frequency shifts. No sta- prewhitening procedure, applied to the light curve of KIC005941844.
tistically significant difference is however found between the ro- The BIC values are relevant for strategies 3, 4, and 5, whereas the
tation rates of 26 of the high- and low- fsv stars, derived by Ped- S/N value and the estimated amplitude A expressed in estimated stan-
ersen et al. (2021), as is illustrated by the box plot in Fig. 5. dard deviations σ̂A are relevant for strategies 1 and 2, respectively.
Weighted averages, on the other hand, indicate a difference, as Bottom: Lomb-Scargle periodogram (in orange) obtained after sub-
illustrated in Tables 2 and 3. The rotation rate for the high- fsv tracting the light curve model obtained by prewhitening strategy 3 for
star we discussed in the previous subsection is located around the KIC005941844, from the original light curve. The extracted frequencies
centre of the respective class rotation rate distributions, whereas for this model are displayed as grey lines with height equal to the ex-
tracted amplitude. Note the different y-axis scale for the inset compared
to the main panel. Article number, page 7 of 16A&A proofs: manuscript no. main
Fig. 4. Distribution of the amplitudes and frequencies of (independent) signals not identified as combination frequencies for all 38 SPB stars in our
sample, for each of the prewhitening strategies defined in Table 1. The colour indicates if these signals are involved in a candidate resonance (‘red’)
or not (‘blue’), and its gradient indicates the number of detected frequencies: colours are brighter if more signals are detected within a specific
amplitude-frequency bin. The ‘red and blue distributions’ are plotted independently, so that overlap between these distributions is the cause for
the appearance of darker, maroon-like coloured bins belonging to the ‘red distribution’. The maximal frequency displayed in each panel is 8 d−1 ,
which is the highest frequency of any of the detected (robust two-signal) candidate resonances (see Section 4, and especially, Fig. 10). We display
independent signals with an amplitude > 10 ppm because very few independent signals have amplitudes smaller than this limit.
the rotation rates of the the low- fsv stars we discuss in this sub- 3.2.1. High mode density stars
section are at the lower boundaries of their distributions (see Ta-
bles 2 and 3 for the explicit values). The rotation rate of the high An example of a high mode density star, a low- fsv star that
mode density star example, KIC008255796, is in fact an outlier does not display outburst-like features in its light curve, is
in the rotation rate box-and-whisker plot for the low- fsv class, KIC008255796. It was first identified as a misclassified B star
next to the slowly-rotating SPB KIC008459899, which is sus- by Zhang et al. (2018). Its light curve is shown in the top panel
pected to be a double lined spectroscopic binary due to slightly of Fig. 6 and displays periodic variations on longer time scales
larger O-C residuals of the fit to its observed spectrum (Lehmann than KIC005941844, the high- fsv star example. The characteris-
et al. 2011). The latter star is the only (suspected) binary within tic properties of the light curves of high mode density stars such
the pseudoclass of low- fsv stars. Whether and how its binary na- as KIC008255796 are not dissimilar to those of high- fsv stars.
ture is correlated with its low rotation rate and low fsv is un- The lower fsv values attained by high mode density stars can be
known. With the exception of this binary and KIC008255796, attributed to the fact that on average maximal excursions from
the low fsv values of the low- fsv stars thus seem to be connected the baseline brightness are smaller, so that small-amplitude unre-
to rapid rotation. solved signals have a larger impact on the fsv value. The maximal
excursion from the light curve baseline for KIC005941844 is
∼ 35 ppt, whereas this value is only ∼ 6 ppt for KIC008255796,
as can be deduced from the top panels of Fig. 3 and Fig. 6.
Article number, page 8 of 16J. Van Beeck et al.: Nonlinear g-mode resonances in SPB stars
Table 2. Rotation rates obtained from averaging over eight theories of
core-boundary and envelope mixing by Pedersen et al. (2021) for the 19
high- fsv stars, if applicable. The prewhitening strategy that delivers the
highest- fsv light curve regression model among the models generated
by all five strategies, is noted in the ‘Strategy’ column. The (approx-
imate) amplitude of the largest-amplitude unresolved signal in the LS
periodogram of the light curve subtracted by the highest- fsv regression
model is noted in the Aunres/res column, where it is symbolically divided
by the amplitude of the largest-amplitude resolved signal in the LS pe-
riodogram of the original light curve.
KIC Ωrot /Ωcrit (%) Strategy Aunres/res (ppm)
003459297 55 ± 6 2 ∼ 135/4600
003839930 / 2 ∼ 330/10000
003865742 88 ± 10 2 ∼ 160/8750
004930889 60 ± 12 3 ∼ 310/8000
004936089 16 ± 7 3 ∼ 80/5700
Fig. 5. Box-and-whisker plot of the rotation rates of 26 SPB stars ob-
005084439 / 2 ∼ 175/3650
tained from averaging over 8 theories of core-boundary and envelope
mixing by Pedersen et al. (2021). These are grouped into the two classes 005309849 46 ± 14 3 ∼ 65/3400
defined in this work: low and high fsv , as indicated schematically on the 005941844 39 ± 4 3 ∼ 300/14000
x-axis. Whiskers are extended past the quartiles by maximally 1.5 times
the interquartile range. A single outlier for the low fsv class is indicated 006352430 50 ± 11 2 ∼ 350/7400
with a rhombus mark. Individual rotation rates and estimated standard 006780397 70 ± 24 3 ∼ 26/440
deviations are indicated in grey. The horizontal offset for these rota- 007760680 25 ± 5 5 ∼ 390/9800
tion rates was imposed for reasons of visibility without any additional
meaning. 008324482 / 2 ∼ 175/2000
008714886 60 ± 18 2 ∼ 155/2150
008766405 77 ± 10 3 ∼ 110/1750
The iteration progress curves of high mode density stars are
009020774 51 ± 10 2 ∼ 75/7700
also not dissimilar to those of high- fsv stars, as is illustrated
by comparing the middle panels of Figs 3 and 6, obtained for 009227988 / 3 ∼ 170/2200
KIC005941844 and KIC008255796, respectively. The strategy 010220209 / 2 ∼ 44/1300
2 and 3 iteration progress curves of KIC008255796 indicate 010526294 4±3 3 ∼ 2000/14500
that more frequencies need to be extracted to reach the ‘optimal
010658302 / 3 ∼ 620/3000
model’ for that strategy. This is not unexpected, because these
strategies focus their extraction efforts on high mode density re- Average 29 a N.A. 300/5807
gions of the periodogram, which are strongly present in the LS
periodograms of high mode density stars. The other strategies Notes. Known binary stars (Pápics et al. 2013, 2017) have their KIC
have triggered their stop criteria, and those in which hinting is identifiers marked in boldface.
S/N-based are especially strict, as expected. (a)
Average weighted rotation rate, where the inverse variances
The eponymous high mode density within the peri- (1/σ2Ωrot /Ωcrit ) are used as weights.
odogram is well-illustrated in the bottom panel of Fig. 6 for
KIC008255796. Higher mode densities cause more signals to
be unresolved with respect to the LD78 criterion, and therefore photometry), remains to be validated. What is clear, however, is
increase the probability that a larger-amplitude signal is unre- that the amplitude modulation observed in some high mode den-
solved. Combined with the on average smaller amplitude signals sity stars could originate from beating of close frequency pairs
observed in high mode density stars compared to high- fsv stars, (Bowman et al. 2016), or nonlinear mode interactions in the ‘in-
which are a corollary of the observed differences in maximal ex- termediate’ regime (e.g., Buchler et al. 1997; Goupil et al. 1998).
cursions from the light curve baseline, a larger influence of unre- Hence, some high mode density stars also are prime targets for
solved signals on fsv is expected. Indeed, as can be derived from in-depth nonlinear asteroseismic analysis.
Tables 2 and 3, the average unresolved/resolved amplitude for
high mode density stars is ∼ 186/750 ppm, whereas that value is
3.2.2. Outbursting stars
∼ 300/5807 ppm for high- fsv stars.
The similarities noted in the light curve and Fourier proper- An example of a low- fsv star that displays the eponymous
ties of high mode density low- fsv stars and high- fsv stars render outburst-like features in its light curve, an outbursting star, is
their classification somewhat artificial. This is why we refer to KIC011971405. It is considered to be a prototypical oscillating
the high- and low- fsv stars as pseudoclasses. It is however ex- Be star by Kurtz et al. (2015), and was first identified by McNa-
pected that amplitude modulation (and frequency modulation) mara et al. (2012) as an SPB oscillator. Pápics et al. (2017) spec-
more frequently occurs for high mode density stars, as they on troscopically confirmed it as a Be star, and studied its variability
average exhibit more unresolved signals. Whether that implies and outbursts in detail. The days-long outburst-like features in its
that the observed differences in fsv between the two pseudo- light curve are readily visible in the upper panel of Fig. 7. Similar
classes are not of a purely instrumental origin (i.e. not being able features can be observed in the light curves of the other outburst-
to resolve high mode densities due to limited time span of the ing stars, with differing lengths and amplitudes. However, as il-
Article number, page 9 of 16A&A proofs: manuscript no. main
Table 3. Same as Table 2, but for the 19 low- fsv stars in our sample.
KIC Ωrot /Ωcrit (%) Strategy Aunres/res (ppm)
003240411 67 ± 5 3 ∼ 500/570
003756031 / 3 ∼ 130/690
004939281* 78 ± 17 3 ∼ 350/1850
006462033 85 ± 6 2 ∼ 60/600
007630417 66 ± 16 2 ∼ 160/540
008057661 59 ± 16 3 ∼ 50/310
008087269 / 3 ∼ 175/1150
008255796 3±5 3 ∼ 280/970
008381949 76 ± 6 3 ∼ 135/650
008459899 8±4 2 ∼ 145/640
009715425* 83 ± 9 1 ∼ 2400/4700
009964614 / 2 ∼ 130/520
×105
010285114* / 3 ∼ 380/730 9.6 Prewhitening 30
010536147 67 ± 8 3 ∼ 145/2075 strategy
1 25
011152422* / 3 ∼ 135/540 9.4 2 25
011360704* 96 ± 4 3 ∼ 1225/2075 3
9.2 4 20
011454304 / 3 ∼ 350/535 20
5
011971405*
A
72 ± 8 2 ∼ 950/3500 9.0
S/N
BIC
15
A/
012258330 73 ± 9 5 ∼ 155/500 15
8.8 10
Average 57 N.A. 441/1218
10
- Outb. a 90 N.A. 907/2232 8.6
- H.m.d. b 44 N.A. 186/750 5
S/N = 4.0 5
8.4
Notes. A suspected binary star (Lehmann et al. 2011) has its KIC iden- 0 25 50 75 100 125 150 175 200
tifier marked in bold. The weighted averages for the rotation rates are
computed in the same way as in Table 2. The numbers in the ‘Strategy’
Prewhitening step
column denote the strategy for which the highest fsv value is obtained. /2 ( Hz)
(*) 0 50 100 150 200 250
Outbursting star. (a) Average sample properties for the outbursting 1000
stars. (b) Average sample properties for the high mode density stars. /2 ( Hz)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5
400
800
lustrated in the same panel of Fig. 7, the parts of the light curve of 300
A (ppm)
KIC011971405 away from these outburst-like features are very
A (ppm)
similar to that of a non-outbursting SPB oscillator. This similar- 600 200
ity is also noted for the other outbursting stars, and fits within
the angular momentum transport model put forward by Neiner 100
400
et al. (2020) that is used to describe Be star outbursts. Within this 0
model gravito-inertial modes stochastically excited by core con- 0.00 0.25 0.50 0.75 1.00 1.25 1.50
vection dominate the LS periodogram at the time of outbursts, 200 /2 (d 1)
whereas g modes excited by the κ mechanism prevail for the en-
tire time span of the light curve. Hence, amplitude and frequency
0
modulation is expected for all outbursting stars, which gives rise 0 5 10 15 20 25
to high mode densities in the periodogram. For KIC011971405 /2 (d 1)
modulation has been detected and characterised by Pápics et al.
(2017). Interestingly, while for five of six outbursting stars these Fig. 6. Same as Fig. 3, but for high mode density star KIC008255796.
high mode densities are distinctly grouped, KIC011152422 dis-
plays a high mode density in its LS periodogram that spans from
0 to 1 d−1 . smallest for all outbursting stars. The stop criteria of strategies 4
The S/N stop criteria trigger an early stop to the prewhiten- and 5 trigger early abortion of the prewhitening for all outburst-
ing (i.e. before 200 frequencies are extracted) for almost all ing stars, whereas the iteration progress curves for strategies 2
outbursting stars, an example of which can be observed for and 3 indicate more frequencies can be extracted.
KIC01197405 in the middle panel of Fig. 7. High mode den- The high mode densities in the LS periodograms and the
sities are the main cause for these early triggers. The exception outburst-like features in the light curves are also the prime fac-
is KIC010285114, whose outburst amplitude, in comparison to tors to which the low fsv values for outbursting stars can be at-
the typical excursions from the baseline in its light curve, is the tributed. Because of these high mode densities we do not ex-
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